Search: id:A110560
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%I A110560
%S A110560 1,3,3,5,5,7,7,1,1,11,11,13,13,1,1,17,17,19,19,1,1,23,23,1,1,1,1,29,29,
%T A110560 31,31,1,1,1,1,37,37,1,1,41,41,43,43,1,1,47,47,1,1,1,1,53,53,1,1,1,1,59,
%U A110560 59,61,61,1,1,1,1,67,67,1,1,71,71,73,73,1,1,1,1,79,79,1,1,83,83
%N A110560 Numerators of T(n+1)/n! reduced to lowest terms, where T(n) are the triangular 
               numbers A000217.
%C A110560 The exponential generating function of the triangular numbers ws given 
               in Sloane & Plouffe as g(x) = (1 + 2x + (x^2)/2)*e^x = 1 + 3*x + 
               3*x^2 + (5/3)*x^3 + (5/8)*x^4 + (7/40)*x^5 + (1/896)*x^6 + (11/72576)*x^7 
               + ... = 1 + 3*x/1! + 6*(x^2)/2! + 10*(x^3)/3! + 15*(x^4)/4! + ...
%D A110560 Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. 
               San Diego, CA: Academic Press, 1995, p. 9.
%H A110560 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               TriangularNumber.html">Triangular Number</a>.
%F A110560 A110560(n)/A110561(n) is the n-th coefficient of the exponential generating 
               function of T(n), the triangular numbers A000217.
%e A110560 a(3) = 5 because T(3+1)/3! = T(4)/3! = (4*5/2)/(1*2*3) = 10/6 = 5/3 so 
               the fraction has numerator 5 and denominator A110561(3) = 3. Furthermore, 
               the 3rd term of the exponential generating function of the triangular 
               numbers is (5/3)*x^3.
%t A110560 T[n_] := n*(n + 1)/2; Table[Numerator[T[n + 1]/n! ], {n, 0, 82}]
%Y A110560 Denominator = A110561.
%Y A110560 Closely related to this is T(n)/n! which is A090585/A090586.
%Y A110560 Sequence in context: A021302 A004649 A002374 this_sequence A141424 A069902 
               A085779
%Y A110560 Adjacent sequences: A110557 A110558 A110559 this_sequence A110561 A110562 
               A110563
%K A110560 easy,frac,nonn
%O A110560 0,2
%A A110560 Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 27 2005
%E A110560 Extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Jul 27 2005

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